Title: On the structure of cofree Hopf algebras

Abstract: We prove an analogue of the Poincaré-Birkhoff-Witt theorem and
of the Milnor-Moore theorem for non-cocommutative Hopf algebras. The
primitive part of a cofree Hopf algebra is a B-infinity-algebra. We
construct a universal enveloping functor U2 from B-infinity-algebras to 2-
associative algebras, i.e. algebras equipped with two associative
operations. We show that any cofree Hopf algebra H is of the form U2(Prim H).